Norman L.J.,Kotz S.,,Balakrishnan N. Continuous univariate distributions, vol. 1 ......Page 4
Copyright ......Page 5
Contents ......Page 7
Preface xv ......Page 14
List of Tables xvii 1......Page 16
TABLE 12.1 Values of A giving maximum or minimum b2 43 ......Page 63
TABLE 13.1 Percentile points of normal distribution, as standardized deviates (values of Ua) 84 ......Page 104
TABLE 13.2 Standardized percentile points of distribution (13.42) and the normal distribution 112 ......Page 132
TABLE 13.3 Efficiency of trimmed means, relative to X 124 ......Page 144
TABLE 13.4 Multipliers an, a'n such that E[anS] = a = E[a'JV] 128 ......Page 148
TABLE 13.5 Values of v, A, and log c such that first three moments of (T/a)x and agree 130 ......Page 150
TABLE 13.7 Relative efficiencies of unbiased estimators of a 136 ......Page 156
TABLE 13.8 Unbiased estimators of a 137 ......Page 157
TABLE 13.9 Efficiency (%) of Winsorized mean based on Xj+1,..., X'n_j relative to the best linear unbiased estimator of f based on X], Xj+l9 ...,X'n_j 149 ......Page 169
TABLE 13.10 Expected value, standard deviation, and (mean deviation)/(standard deviation) for truncated normal distributions 159 ......Page 179
TABLE 13.11 Asymptotic values of n X (variance)/cr2 for maximum likelihood estimators 160 ......Page 180
TABLE 14.1 Standardized 100a% points (x'a) of lognormal distributions (a = §_1)\t214......Page 234
TABLE 14.2 Values of a3 and a4 for distribution (14.2y\t215......Page 235
TABLE 14.3 Percentile points of standardized lognormal distributions 223......Page 243
TABLE 14.4 Values of -u1/(n + 1) and -E(Ul n) 231......Page 251
TABLE 14.6 Maximum likelihood estimators for censored three-parameter lognormal distribution where value of 0 is supposed known 242......Page 262
TABLE 16.2 Percentage points of standard Cauchy distribution 300......Page 320
7.4 Order Statistics and Estimators Based on Order Statistics, 370 ......Page 390
TABLE 17.2 Details of available tables on moments of gamma order statistics 371......Page 391
TABLE 17.4 Values of Pr[Ur s > u] forn = 5, r = 1, 5 = 2 373......Page 393
TABLE 18.1 Comparison of approximations to x2 percentile points 427......Page 447
TABLE 18.2 Accuracy of several chi-squared approximation formulas 434......Page 454
TABLE 18.3 Maximum absolute error in approximations of Fxi(x) in (18.22), (18.23), (18.24), and (18.51) 438......Page 458
6 Characterizations, 441 ......Page 461
8 Distributions of Linear Combinations, 444 ......Page 464
TABLE 18.6 Comparison of exact percentage points andapproximations for Qk\t446......Page 466
TABLE 18.7 Percentage points using Imhof s method (I),Solomon and Stephens’ method (S), and mixture approximation (M) 449......Page 469
TABLE 18.8 Coefficients at of the BLUE of a and efficiencyrelative to the BLUE based on complete sample for n = 2(1)10, r = 0, and s = 0(1)« — 2 464......Page 484
TABLE 18.9 Optimum spacings {A,}, corresponding coefficientsbt, and the value of K2 for the ABLUE cr** (complete samples) 466......Page 486
TABLE 19.1 A dictionary of characterizations of the exponentialdistribution 535......Page 555
TABLE 20.1 Values of for specific estimator of\tx^ ofthe form (20.64) 591......Page 611
5.9 Censored Data, 592 ......Page 612
TABLE 21.1 Moments of Weibull distributions 633......Page 653
TABLE 21.2 Values of gn = log10 w1/2 such thatgn(log10 X'n - log10 X()-1 is a median-unbiased estimator of c 646......Page 666
1 Introduction, 1 ......Page 21
2 Order Statistics, 6 ......Page 26
3 Calculus of Probability Density Functions, 14 ......Page 34
4.1 Pearson System, 15 ......Page 35
4.2 Expansions, 25 ......Page 45
4.3 Transformed Distributions, 33 ......Page 53
4.4 Bessel Function Distributions, 50 ......Page 70
4.5 Miscellaneous, 53 ......Page 73
5 Cornish-Fisher Expansions, 63 ......Page 83
6 Note on Characterizations, 67 ......Page 87
Bibliography, 68 ......Page 88
1 Definition and Tables, 80 ......Page 100
2 Historical Remarks, 85 ......Page 105
3 Moments and Other Properties, 88 ......Page 108
4 Order Statistics, 93 ......Page 113
5 Record Values, 99 ......Page 119
6 Characterizations, 100 ......Page 120
7 Approximations and Algorithms, 111 ......Page 131
8.1 Estimation of ?, 123 ......Page 143
8.2 Estimation of a, 127 ......Page 147
8.3 Estimation of Functions of ? and a, 139 ......Page 159
8.4 Estimation from Censored Data, 146 ......Page 166
9.1 Box-Muller Method, 152 ......Page 172
9.3 Acceptance-Rejection Method, 153 ......Page 173
9.4 Ahrens-Dieter Method, 155 ......Page 175
10.1 Truncated Normal Distributions, 156 ......Page 176
10.2 Mixtures, 163 ......Page 183
10.3 Other Related Distributions, 168 ......Page 188
Bibliography, 174 ......Page 194
1 Introduction, 207 ......Page 227
2 Historical Remarks, 209 ......Page 229
3 Moments and Other Properties, 211 ......Page 231
4.1 0 Known, 220 ......Page 240
4.2 0 Unknown, 222 ......Page 242
4.3 Graphical Estimation, 236 ......Page 256
5 Tables and Graphs, 237 ......Page 257
6 Applications, 238 ......Page 258
7 Censoring, Truncated Lognormal and Related Distributions, 240 ......Page 260
8 Convolution of Normal and Lognormal Distributions, 247 ......Page 267
Bibliography, 249 ......Page 269
1 Introduction, 259 ......Page 279
2 Genesis, 260 ......Page 280
3 Definition, 261 ......Page 281
4 Moments, 262 ......Page 282
5 Properties, 266 ......Page 286
6 Estimation of Parameters, 270 ......Page 290
7 Truncated Distributions—Estimation of Parameters, 276 ......Page 296
7.1 Doubly Truncated Distribution, 277 ......Page 297
7.2 Truncation of the Lower Tail Only, 278 ......Page 298
8 Conditional Expectations of the Estimators of the Cumulants, 279 ......Page 299
9.1 Reciprocal of an Inverse Gaussian Variate, 281 ......Page 301
9.2 Birnbaum-Saunders Distribution, 283 ......Page 303
9.3 Generalized Inverse Gaussian Distributions, 284 ......Page 304
9.4 Mixtures of IGK/i, A) with Its Complementary Reciprocal, 285 ......Page 305
9.5 Other Related Distributions, 287 ......Page 307
10 Tables, 289 ......Page 309
11 Applications, 290 ......Page 310
Bibliography, 292 ......Page 312
1 Historical Remarks, 298 ......Page 318
2 Definition and Properties, 299 ......Page 319
3 Order Statistics, 303 ......Page 323
4.1 Methods Based on Order Statistics, 306 ......Page 326
4.2 Maximum Likelihood Inference, 310 ......Page 330
4.3 Conditional Inference, 314 ......Page 334
4.4 Bayesian Inference, 315 ......Page 335
4.5 Other Developments in Inference, 317 ......Page 337
5 Genesis and Applications, 318 ......Page 338
6 Characterizations, 321 ......Page 341
7.1 Monahan’s (1979) Algorithm, 323 ......Page 343
7.2 Kronmal and Peterson’s (1981) Acceptance-Complement Method, 324 ......Page 344
7.3 Ahrens and Dieter’s (1988) Algorithm, 326 ......Page 346
8 Related Distributions, 327 ......Page 347
Bibliography, 329 ......Page 349
1 Definition, 337 ......Page 357
2 Moments and Other Properties, 338 ......Page 358
3 Genesis and Applications, 343 ......Page 363
4 Tables and Computational Algorithms, 344 ......Page 364
5 Approximation and Generation of Gamma Random Variables, 346 ......Page 366
6 Characterizations, 349 ......Page 369
7 Estimation, 355 ......Page 375
7.1 Three Parameters Unknown, 356 ......Page 376
7.2 Some Parameters Unknown, 360 ......Page 380
7.3 Estimation of Shape Parameter and y Known), 368 ......Page 388
8 Related Distributions, 379 ......Page 399
8.1 Truncated Gamma Distributions, 380 ......Page 400
8.2 Compound Gamma Distributions, 381 ......Page 401
8.3 Transformed Gamma Distributions, 382 ......Page 402
8.4 Convolutions of Gamma Distributions, 384 ......Page 404
8.6 Reflected Gamma Distributions, 386 ......Page 406
8.7 Generalized Gamma Distributions, 388 ......Page 408
Bibliography, 397 ......Page 417
1 Historical Remarks, 415 ......Page 435
2 Definition, 416 ......Page 436
3 Moments and Other Properties, 420 ......Page 440
4 Tables and Nomograms, 422 ......Page 442
5 Approximations and Computational Algorithms, 426 ......Page 446
7 Simulational Algorithms, 443 ......Page 463
9 Related Distributions, 450 ......Page 470
10.2 Basic Properties, 456 ......Page 476
10.3 Order Statistics and Properties, 459 ......Page 479
10.4 Inference, 461 ......Page 481
10.5 Prediction, 474 ......Page 494
10.6 Record Values and Related Issues, 475 ......Page 495
10.7 Related Distributions, 479 ......Page 499
Bibliography, 481 ......Page 501
2 Genesis, 494 ......Page 514
3 Some Remarks on History, 497 ......Page 517
4 Moments and Generating Functions, 498 ......Page 518
6 Order Statistics, 499 ......Page 519
7.1 Classical Estimation, 506 ......Page 526
7.2 Grouped Data, 509 ......Page 529
7.3 Estimators Using Selected Quantiles, 510 ......Page 530
7.4 Estimation of Quantiles, 521 ......Page 541
7.5 Bayesian Estimation, 522 ......Page 542
7.6 Miscellaneous, 526 ......Page 546
8 Characterizations, 534 ......Page 554
8.1 Characterizations Based on Lack of Memory and on Distributions of Order Statistics, 536 ......Page 556
8.2 Characterizations Based on Conditional Expectations (Regression), 540 ......Page 560
8.4 Miscellaneous, 544 ......Page 564
8.5 Stability, 545 ......Page 565
9 Mixtures of Exponential Distributions, 546 ......Page 566
10 Related Distributions, 551 ......Page 571
Bibliography, 556 ......Page 576
2 Genesis, 573 ......Page 593
3 Definitions, 574 ......Page 594
4.2 Alternative Measures of Location, 577 ......Page 597
4.3 Measures of Inequality, 578 ......Page 598
5 Estimation of Parameters, 579 ......Page 599
5.2 Estimators from Moments, 580 ......Page 600
5.3 Maximum Likelihood Estimation, 581 ......Page 601
5.4 Estimation Based on Order Statistics, 584 ......Page 604
5.6 Minimax Estimation, 588 ......Page 608
5.7 Estimation of Pareto Densities, 589 ......Page 609
5.8 Estimation of Pareto Quantiles, 590 ......Page 610
5.10 Bayesian Estimation, 594 ......Page 614
6 Estimation of Lorenz Curve and Gini Index, 595 ......Page 615
7 Miscellaneous, 596 ......Page 616
8.1 Order Statistics, 599 ......Page 619
8.2 Record Values, 601 ......Page 621
9 Characterizations, 603 ......Page 623
10 Product and Ratios of Pareto Random Variables, 605 ......Page 625
11 Applications and Related Distributions, 607 ......Page 627
12 Generalized Pareto Distributions, 614 ......Page 634
Bibliography, 620 ......Page 640
1 Historical Remarks, 628 ......Page 648
2 Definition, 629 ......Page 649
3 Order Statistics, 637 ......Page 657
4.1 Moment Estimation, 641 ......Page 661
4.2 Best Linear Unbiased Estimation, 644 ......Page 664
4.3 Asymptotic Best Linear Unbiased Estimation, 647 ......Page 667
4.4 Minimum Quantile Distance Estimation, 651 ......Page 671
4.5 Modified Moment Estimation, 652 ......Page 672
4.6 Maximum Likelihood Estimation, 656 ......Page 676
4.7 Modified Maximum Likelihood Estimation, 660 ......Page 680
4.8 Bayesian Estimation and Shrinkage Estimation, 661 ......Page 681
5 Tolerance Limits and Intervals, 663 ......Page 683
6 Prediction Limits and Intervals, 667 ......Page 687
7 Record Values, 671 ......Page 691
8 Tables and Graphs, 675 ......Page 695
9 Characterizations, 679 ......Page 699
10 Simulation Algorithms, 682 ......Page 702
11 Applications, 684 ......Page 704
12 Related Distributions, 686 ......Page 706
Bibliography, 695 ......Page 715
Abbreviations, 723 ......Page 743
Author Index, 725 ......Page 745
Subject Index, 749 ......Page 769
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